Problem 13 - Technical Mathematics with Calculus

Question

Solve each problem for the required quantity. The sum of 45 and some number equals 6 times that number. Find it.

Step-by-Step Solution

Answer

The unknown number is 9.

Step 1: Set up the equation

Let the unknown number be represented by the variable 'n'. According to the problem, the sum of 45 and the number equals 6 times the number. The equation therefore is: 45 + n = 6n.

Step 2: Isolate the variable

To find the value of 'n', we need to isolate 'n' on one side of the equation. This can be done by subtracting 'n' from both sides of the equation which gives us: 45 = 6n - n.

Step 3: Simplify the equation

Simplify the right side of the equation by combining like terms: 45 = 5n.

Step 4: Solve for the unknown number

To solve for 'n', divide both sides of the equation by 5. This gives us: n = 45 / 5.

Step 5: Find the value of the unknown number

Divide 45 by 5 to get the value of 'n': n = 9.

Key Concepts

Isolating Variables

Understanding how to isolate a variable is a cornerstone of solving algebraic equations. It involves manipulating the equation so that the variable we want to solve for, also known as the unknown, stands alone on one side of the equality sign. This process typically involves a sequence of operations that cancels out other terms and leaves the variable by itself.

For example, consider an equation like \(45 + n = 6n\). To isolate \(n\), we subtract \(n\) from both sides to balance the equation, resulting in \(45 = 5n\). The goal is to perform operations that do not change the equality but simplify the equation to make the variable apparent and easy to solve. This process, when executed properly, guides us to the solution systematically.

Solving for Unknown Numbers

When students are tasked with solving equations to find an unknown number, the approach often requires a strategic combination of algebraic techniques, including isolation of variables and balancing operations. The ultimate objective is to determine the value of an unknown quantity that makes the equation true.

In the given exercise, we proceed from \(45 = 5n\) to find \(n\) by performing division, one of the fundamental operations, on both sides of the equation. By dividing by the coefficient of the variable (5 in this case), we obtain \(n = 45 / 5\), which simplifies to \(n = 9\). This solution reflects the number which, when replaced in the original equation, maintains the equation's balance.

Combining Like Terms

Combining like terms is an essential procedure to simplify algebraic expressions and equations. 'Like terms' are terms that contain the same variables raised to the same power. By combining them, we streamline the equation and make it easier to solve for the unknown.

In the exercise presented, we encounter the equation \(45 = 6n - n\). Here, both terms on the right side of the equation involve 'n'. Since they are like terms, we combine them by addition or subtraction, which, in this case, results in \(5n\). The equation now reads \(45 = 5n\), which is an easier version of the original equation, allowing us to swiftly isolate \(n\) and solve for the unknown number.